Prob #1
with(plots):
Diag:=proc(a,b) local i,P,Q,R,PQ,curve,Points,Labels:
P:=RP(a,b):
Q:=RP(a,b):


R:=simplify(AE(a,b,P,P)):
PQ:=simplify(AE(a,b,R,R)):
 
curve:=implicitplot(y^2=x^3+a*x+b): 
Points:=plot([P,Q,R,PQ], style=point, color=blue, symbol=solid circle, symbol size=12): 
Labels:=textplot([[P[],"A"],[Q[],"B"],[R[],"C"],[PQ[],"A+B"]], font=[times,16], align={left,above}): 
display(curve,Points,Labels):
end:


Prob #2
IntSol:=proc(a,b,K) local x,y,S:
S:={}:
 for x from -K to K do
  y:=sqrt(x^3+a*x+b):
  if type(y,integer) then
    S:=S union {[x,y]}:
  fi:
 od:
S:
end:


Test run
a:=1: b:=2: IntSol(a,b,10000);
                       {[-1, 0], [1, 2]}
a:=2: b:=2: IntSol(a,b,10000);
                               {}
a:=2: b:=3: IntSol(a,b,10000);
                       {[-1, 0], [3, 6]}


Prob #3
SolChain:=proc(a,b,S1,S2,K) local L,i:
L:=[S1,S2]:
for i from 1 to K do
 L := [op(L), AddE(a,b,L[-2],L[-1])]:
od:
L: 
end: 


Test run
Cannot get infinite solutions this way
a:=2: b:=3: P,Q := op(IntSol(a,b,1000)); L1:=SolChain(a,b,P,Q,10); L2:=SolChain(a,b,P,Q,100); L1 minus L2;
                    P, Q := [-1, 0], [3, 6]


                /                 [-193101  -53536482]  [1  -15]  
   L1 := { [-1, 0], [3, 6], [-------, ---------],          [-, ---], 
                \                 [207025   94196375 ]  [4   8 ]  


     [599  -15420]\ 
     [---, ------] }
     [121   1331 ]/ 


          /                       [-193101  -53536482]  [1  -15]  
   L2 := { [-1, 0], [3, 6], [-------, ---------],          [-, ---], 
          \                       [207025   94196375 ]  [4   8 ]  


     [599  -15420]\ 
     [---, ------] }
     [121   1331 ]/ 
                               {}


a:=2: b:=3: P:=RP(a,b); Q:=RP(a,b); L1:=SolChain(a,b,P,Q,20): L2:=SolChain(a,b,P,Q,30): L1 minus L2;
                         [81    5151441^ (1/2)]
                 P := [---, ----     ]
                        [100  1000             ]
                        [183   3 (1420943)^ (1/2)]
                 Q := [---   ,   ----     ]
                         [100  1000             ]


                               {}